Chapter 4. Quadratic Equations   

Chapter 4. Quadratic Equations

Exercise 4.1 complete solution Exercise 4.2 complete solution Exercise 4.3 complete solution Exercise 4.4 complete solution

Important notes : 1. A quadratic equation in the variable  is an equation of the form , where  are real numbers and  . For example :  etc . 2. Two types of the quadratic equation are : Pure quadratic equation :  The general form of the pure quadratic equation is  , where are real numbers and  . For example :  ,………. , etc . Complete quadratic equation : The general form of the complete quadratic equation is  , where are real numbers and  . For example :  ,………. , etc . 3. If  and  are the roots of the quadratic equation  , then The sum of the roots The product of the roots 4. The method of completing the square : Let , the quadratic equation  .        or  5. Quadratic Formula :   The roots of a quadratic equation  are given by , provided  . 6. The discrimanant of the quadratic equation  is given by  . 7. A quadratic equation  has   (i) two distinct real roots, if   ,   (ii) two equal real(i.e., coincident) roots, if   ,   (iii) no real roots, if   .   (iv) If , then the roots are reciprocal to each other .

Class 10 Maths  Chapter 4. Quadratic Equations Exercise 4.1 Solutions

1.Check whether the following are quadratic equations :

 (i)        

(ii)      

(iii) 

(iv)    

(v)   

(vi)  

(vii)      

(viii)  

Solution : (i)       

 

 

 

It is of the form  

 So, the given equation is a quadratic equation .

 (ii)      

 

   

   

It is of the form  

So, the given equation is a quadratic equation .

(iii) 

 

 

 

 

It is not of the form  

So, the given equation is not  a quadratic equation .

(iv)    

 

 

 

It is of the form  

So, the given equation is a quadratic equation .

(v)   

 

 

 

 

It is of the form  

So, the given equation is a quadratic equation .

(vi)  

 

 

 

It is of the form

So, the given equation is a quadratic equation .

(vii)     

 

 

 

It is not of the form  

So, the given equation is not a quadratic equation .

(viii)   

 

 

 

 

It is of the form  

So, the given equation is a quadratic equation .

2. Represent the following situations in the form of quadratic equations :

(i) The area of a rectangular plot is 528  . The length of the plot (in metres) is one more than twice its breadth . We needed to find the length and breadth of the plot .

Solution :  Let  be the breadth of the plot  and the length of the plot will be  .(in metres)

   A/Q ,  

 

Therefore , the given equation is a quadratic equation .

We have,

 or 

          

(Impossible )

Therefore, the breadth of the plot is 16 m and the length of the plot is 2 ×16 +1 = 33 m

(ii) The product of two consecutive positive integers is 306 . We need to find the integers .

Solution :   Let  and  are two consecutive positive integers respectively .

 A/Q, 

  

Therefore, the  given equation is a quadratic equation .

We have,

       or   

               

(Impossible)

Therefore, the two consecutive positive integers 17 and 18 (= 17 + 1) .

(iii) Rohan’s mother is 26 years older than him . The product of their ages (in years) 3 years from now will be 360 .We would like to find Rohan’s present age .

Solution :  Let  be the present age of Rohan and Rohan’s mother age will be  years .

 Again , 3 years from now , the age of Rohan and Rohan’s mother will be and  years respectively .   

A/Q, 

   

   

    

     

Therefore, the  given equation is a quadratic equation .

We have,

    or  

  or                       

(impossible)

Therefore, the present age of Rohan is 7 years .

(iv) A train travels a distance of 480 km at a uniform speed . If the speed had been 8 km/h less , then it would have taken 3 hours more to cover the same distance . We need to find the speed of the train .

 Solution :  Let  (in km/hrs) be the speed of the train .

A/Q, 

 

 

Therefore, the  given equation is a quadratic equation .

We have ,

   or 

 or  

(Impossible) 

Therefore, the speed of the train is 32 Km/h .

Class 10 Maths  Chapter 4. Quadratic Equations Exercise 4.2 Solutions :

1. Find the roots of the following quadratic equations by factorization :

(i)     

Solution : We have,

        

 

 

 

So,    

or    

Therefore, the roots are 5 and  .

(ii)    

Solution : We have ,

          

 

 

So,     

or  

Therefore, the roots are   and   .

(iii) 

Solution : we have , 

       

  

  

   

  So,   

or 

Therefore , the roots are  and  .

(iv)  

Solution : We have ,

    

    

    

     

   So,

or  

Therefore, the roots are  and    .

(v)   

Solution : We have , 

    

 

 

 

or   

Therefore, the roots of   and   .

(vi) 

Solution: We have,

  or 

        or    

                           

Therefore, the roots are 2 and .

(vii) 

Solution: We have,

 

Or 

 

Therefore, the roots are – 6 and 16 .

(viii) 

Solution : We have,

 

Or 

 

Therefore, the roots are  and .

(ix) 

Solution: We have, 

 

 

Or

 

Therefore, the roots are  and  .

(x) 

Solution : We have,

 

 

Or   

 

 

Therefore, the roots are 5 and  –

2. Solve the problems :

(i) John and Jivanti together have 45 marbles . Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124 . We would like to find out how many marbles they had to start with .

Solution : Let  be the marbles of John and Jivanti’s marbles will be  . If 5 marbles has lost , then John and Jivanti’s marble will be  and  respectively .

A/Q,

 

 

 

 

 

 

 

         or   

 Required the solutions are 9 and 36 .

(ii)  A cottage industry produces a certain number of toys in a day . The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day .On a particular day, the total cost of production was Rs. 750 . We would like to find out the number of toys produced on that day .

Solution : Let  be the number of toys produced on that day and the cost of production of each that day Rs  .

A/Q, 

 

       

or  

So, the number of toys is 25 or 30 .

3. Find two numbers whose sum is 27 and product is 182 .

Solution : Let  be the one number and other number will be  .

A/Q,  

 

 

 

 

 

      

 or    

So, the two number are  13 and 14 .

4. Find two consecutive positive integers, sum of whose squares is 365 .

Solution:  Let  and  are two consecutive positive integers respectively .

  A/Q ,   

 

 

 

 

 

 

     

 or    

So, the two consecutive positive integers are 13 and 14 (= 13+1) .

5. The altitude of a right triangle is 7 cm less than its base . If the hypotenuse is 13 cm , find the other two sides .

Solution : Let  (in cm) be the altitude of a right triangle and the base is  cm .

  A/Q, 

 

 

 

 

 

  

        

    or   (impossible)

So, the two sides of the triangle are 12 cm and 5 cm (12 – 7 = 5) .

6. A cottage industry produces a certain number of pottery articles in a day . If was observed on a particular day  that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day . If the total cost of production on that day was Rs 90 , find the number of articles produced and the cost of each article .

Solution: Let  be the number of articles produced in a day  and the cost of product is Rs  .

A/Q ,

 

 

 

 

      

 or  

  (impossible)

So, the number of articles produced in a day is 6 and the cost of each item is Rs 15  (2×6 + 3 = 15)  .

Class 10 Maths  Chapter 4. Quadratic Equations Exercise 4.3 Solutions   

1. Find the roots of the following quadratic equations, if they exist , by the method of completing the square :

(i)     

Solution : (i) We have ,

       

 Here , 

                         

Now ,

So, the roots are  3  and    .

(ii)    

Solution :  We have, 

          

Here ,

 

 

 

Now ,

Therefore, the roots of given quadratic equation are    and  .

(iii)     

Solution : we have ,

           

Here , 

 

        

        

Now , 

Therefore, the roots of given equation is  and    .

(iv)  

Solution : We have ,

         

 Here , 

 

                            

Since the square of a real number cannot be negative , therefore the quadratic equation has no real value .

2. Find the roots of the quadratic equation by applying the quadratic formula :

(i)    

Solution : We have , 

           

   Here , 

 

                             

We know that ,

 Therefore, the roots of the given equation are 3 and   .

(ii)    

Solution : We have, 

           

 Here ,

 

                             

We know that,

Therefore, the roots of the given equation are   and .

(iii)     

Solution : We have, 

          

   Here , 

 

                             

We know that,

Therefore , the roots of given equation are and .

(iv)  

Solution : We have,

         

   Here , 

 

                              

We know that,

Therefore, the roots of given equation are   and  .

3. Find the roots of the following equations :

(i) 

(ii)  

(iii) 

(iv)

(v) 

(vi) 

Solution :  (i) We have,

 

      

 Here ,  

We know that, 

Therefore, the roots of the given equation are and  .

(ii)  

Solution :  We have , 

 

  

  

   

 

 

   or

              

Therefore , the roots of the equations are 1 and 2 .

Solution : (iii)  We have ,

  or 

  or      

                           

Therefore, the roots are and .

Solution : (iv) We have, 

  or  

  

   

Therefore, the roots are and .

Solution : (v) We have,

  or 

 or  

Therefore, the roots are 1 and 1 .

Solution : (vi)  We have,  

   or

     or  

                      

Therefore, the roots are 3 and .

4. The sum of the reciprocals of Rehman’s ages, (in years) 3 years ago and 5 years from now is  . Find his present age .

Solution : Let  (in years)be present age of Rehman .

3 years ago , the age of Rehman was  years  

and 5 years from now , Rehman age will be  years .

A/Q,

 

 

 

 

 

Or   

 Therefore, the roots of the equation is 7 and – 3 .

5. In a class test, the sum of Shefali’s marks in Mathematics and English is 30 . Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210 . Find her marks in the two subjects .

Solution : Let  be the marks in Mathematics of Shefali and her English marks will be  .

A/Q ,

 

 

 

 

 

 

 

 

       or  

Therefore, the marks obtained by Shefali is 12 or 18 and 13 or 17 respectively .

6. The diagonal of a rectangular field is 60 m more than the shorter side . If the longer side is 30 m more than the shorter side, find the sides of the field .

Solution :  let,  be the shorter side of a rectangular field and the longer side will be  m

 Therefore, the diagonal of a rectangular field is  m .

 A/Q , 

 

           

       

 

       

and   (Impossible)

 Thus, the shorter side of a rectangular field is 90 m and the longer side is  .

7. The difference of square of two numbers is 180 . The square of the smaller number is 8 time the larger number . Find the two numbers .

Solution : Let  and  be two numbers .

A/Q ,   

And   

From  and  , we get

    

 

 

Or    [Impossible]

Putting  in  , we have

    

 

Therefore, the numbers are 18 , 12 and  18 ,  – 12  .

8. A train travels 360 km at a uniform speed . If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey . Find the speed of the train .

Solution : Let  (in km/h) be the speed of the train .

A/Q, 

 

 

 

 

 [Impossible]

or   

Therefore, the speed of the train is 40 km/h .

9. Two water taps together can fill a tank in hours . The tap of larger diameter takes 10 hours less than the smallest one to fill the tank separately . Find the time in which each tap can separately fill the tank .

Solution:  let be the time taken by smaller diameter tap and  be the time taken by larger diameter tap.

A/Q, 

 

 

  

  

Therefore,    or 

 

     [  ]

Therefore, the time taken by smaller diameter tap is 25 hours and  the time taken by larger diameter tap is 15 hours .

10. An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations) . If the average speed of the express train is 11 km/h more than that of the passenger train, find the average speed of the two trains .

Solution : Let  (in km/h) be the speed of the express train and the average speed of the express train is  km/h .

A/Q,   

 

 

 

 

 

 

Or    [Impossible]

Therefore, the speed of the express train is 44 km/h and the average speed of the express train is 33 km/h . [  ] .

11. Sum of the areas of two squares is 468  . If the difference of their perimeter is 24 m , find the sides of the two squares

Solution : Let  and  be the sides of the two squares respectively .

  A/Q , 

 

 And

  [ From  ]

 

 

  [imposible]

or 

Putting the value of  in equation , we get   

Therefore, the sides of the two squares are 18 m and 12 m respectively .

Class 10 Maths  Chapter 4. Quadratic Equations Exercise 4.4 Solutions

1. Find the nature of the roots of the following quadratic equations . If the real roots exists, find them :

  (i)      

Solution : We have ,

                 

 Here ,  

 

                

Therefore, the equation has no real roots .

(ii)      

Solution : We have ,

             

Here ,  

 

                  

Therefore, the roots of the equation are equal .

Using quadratic formula , we have

  or  

   The roots of the equation are and    .

 (iii)    

Solution :  We have ,

              

Here ,  

   

                    

Therefore, the roots of the equation has two distinct real roots .

Using quadratic formula , we have

 

or 

   The roots of the equation are  and    .

2. Find the values of  for each of the following quadratic equations, so that they have two equal roots .

     (i)        

Solution : We have ,        

  Here,  

   

 

 

 

 

 

Therefore, the value of k is  .

  (ii) 

Solution : We have ,

   

 

Here ,  

 

 

   [ impossible] 

or    

Therefore, the value of  is 6 .

(iii) 

Solution : We have , 

Here,   , ,  

A/Q , 

 or – 2

Therefore, the value of k are 2 and  – 2 .

(iv) 

Solution: We have,

Here, 

A/Q,  

 

Therefore, the value of k is – 2  .

(v) 

Solution : We have,

Here,

A/Q, 

Therefore, the value of k is 4 .

(vi)  

Solution : We have, 

Here, 

A/Q,  

  or 

           

(impossible)

Therefore, the value of k is 14 .

3. Is it possible to design a rectangular mango grove whose length is twice its breadth , and the area is 800  ? If so, find its length and breadth .

Solution : Let  and  be the length and breadth of the rectangular mango grove respectively.

A/Q ,

And 

 [ Only positive value]

Putting the value of  in  , we get

        

 

 Yes . So, the length and breadth of rectangular design is 40 m and 20 m respectively .

4. Is the following situation possible ? If so, determine their present ages .The sum of the ages of two friends is 20 years . Four years ago, the product of their ages in years was 48 .

Solution : Let  (in years) be the present age of  one friend and other friend age will be  years .

  Four years ago , the two friend age will be  and  years respectively .

A/Q, 

 

 

 

 

Here,  

 

                

So, the given equation has no real root . Therefore, the given situation is not possible .

5. Is it possible to design a rectangular park of perimeter 80 m and area 400  ? If so, find its length and breadth .

Solution : Let  and  be the length and breadth of the rectangular park .

A/Q, 

 

 

And 

 

  or

 

Putting the value of  in  , we get

     

Yes . So, the length and breadth of the park is 20 m and 20 m respectively .